as far as the physical justification for the constraints, are there any?

The modulus squared of the wave function is a probability distribution. Is there any physical reason to say that at the point [tex]x = 0 + \epsilon [/tex] there's one probability density, and then at [tex]x = 0 - \epsilon[/tex] it's wildly different?

Also, the continuous first derivative rule comes from a similar notion with regards to the momentum. I'll leave it to you to figure that one out.